
where \(h\)  is the Hellinger distance introduced in (REF ). Whenever the regression function \({\gamma }^{*}\)  exists, we automatically deduce a performance of \(\widehat{{\gamma }}_{\widehat{\lambda }}\)  with respect to \({\gamma }^{*}\)  by the distance \(d\left({\gamma }^{*},\widehat{{\gamma }}_{\widehat{\lambda }}\right)={\mathbf {h}}\left({\mathbf {R}}_{{\gamma }^{*}},{\mathbf {R}}_{\widehat{{\gamma }}_{\widehat{\lambda }}}\right)\) . In particular, in the context of changepoint detection problem in exponential families where \(W_{i}\)  are deterministic, the loss function \({\mathbf {h}}\)  is the sum of the Hellinger distances between each two probabilities \(R_{i}\)  and \(R^{\prime }_{i}\) . Such a loss function has been considered in several literature, for instance [1]} and [2]}. From this point of view, unlike the typical methods detecting changes for some parameter of a distribution (for example detecting changes in means for Gaussian and Poisson distributions), our approach validates the changes along the sequence if there are abrupt variations with respect to the distribution level.
